The Design and Performance of an Astigmatic Laser Mode Converter

1. The Design and Performance of an Astigmatic Laser Mode Converter Alex Ellis, Stuyvesant High School Dr. Harold Metcalf and Dr. John Noé, Stony Brook University

2. A What? An astigmatic mode converter is a pair of cylindrical lenses, accompanied by mode-matching lenses, which introduce an astigmatic phase shift to a laser beam. In particular, it can convert Hermite-Gaussian (HG) rectangularly symmetric transverse modes into Laguerre-Gaussian (LG) circularly symmetric modes.

3. Hermite-Gaussian Modes • Rectangularly symmetric • Obtainable in spherical-planar resonator by adjusting plane mirror angle • Also obtainable by an intra-cavity crosswire, when the laser is operating in a multimode • Indices (m,n) refer to the number of intensity minima in the x and y directions HG0,1 HG1,2

4. Laguerre-Gaussian Modes (Optical Vortices) • Circularly Symmetric • Obtained by intra-cavity circular absorbers, computer-generated holograms, or an astigmatic mode converter • Photons have orbital angular momentum of l*hbar • Index l is the number of 2π phase shifts around the diameter LG3,0 LG1,1

5. Superpositions HG0,1+HG1,0 = HG0,1 rotated 45° HG0,1+i*HG1,0 =LG1,0 i represents a π/2 phase shift

6. Operating Principle of an Astigmatic Mode Converter Input The Guoy Phase 45° rotated HG0,1 • HG: (m+n+1)ψ(z) • LG: (l+2p+1)ψ(z) Ψ(z) = arctan(z/z0) Cylindrical Lenses • First introduces path length difference, altering the superposition due to the Guoy phase (astigmatism in the region) • Second one removes the introduced astigmatism

7. Mode Converter Rules Localized Astigmatism and Proper Guoy Phase Shift Mode-Matching A beam waist is required midway between the two cylindrical lenses w0, z0 w0’, z0’

8. Apparatus – The Laser Melles-Griot 05-LHB-570 Helium-Neon tube R=600mm spherical mirror, 295mm from plane mirror output coupler Adjustment of OC gives multimodes, and an intra-cavity crosswire on a translation stage selects particular HG modes

9. Apparatus – The Mode Converter f1 = 300mm fcyl = 150mm d1 = 513mm d2 = 481mm d3 = 212mm w0 = 246um z0 = 300mm w0’ = 227um z0’ = 256mm

10. Achieved Mode Conversions HG0,1 LG1,0 HG0,2 LG2,0 LG3,0 HG0,3 HG1,1 LG1,0 HG1,2 LG1,1

11. Performance – Multimodes CCD Image Voltage in Photodetector (V) 2.06 1.59 2.03 2.22 Optical Intensity (mW) 0.44 0.34 0.43 0.47 Achieved HG Modes (m,n) 01,01,02,03,bad01, bad11,bad1202,03, bad12 01,02,03,bad12 02,bad01 02,bad11,bad12

12. Performance – Mode Compositions d1 d2

13. Performance – Mode Compositions d1 d1 d2 d2 Analysis of this particular mode gives PHG≈ 73% and PLG ≈ 27%

14. Applications • Generation of arbitrary order Bessel beams by use of an axicon • “Optical spanners” that both trap and spin microscopic particles • Further study involving the orbital angular momentum of light

15. Acknowledgements • Dr. Harold Metcalf • Dr. John Noé • Mr. Don Bucher • Mrs. Anne Manwell

16. Further Research • Change of mode converter parameters due to the increased spot size of high-order HG modes • Interferometry of LG beams with plane waves to investigate the spiral phase pattern • Measurement of the l*hbar per photon orbital angular momentum

17. Questions